Coasting-wheel with aeronautic levitation for harmonic electric generation

ABSTRACT

The present invention titled: a coasting-wheel with aeronautic levitation for harmonic electric generation is a device to generate electric energy or power via the principle of electromagnetic induction during the phase of energy-in and during the coasting phase; energy-off. It differs from the conventional electric generators and alternators as such the latter include an element of stator and an element of rotor but in the present invention the armature and magnets undergo rotation. Second, all electric generators operate in non-harmonic mode, i.e., when the input fuel is turned off the electric generator or alternator stops but in the present invention the electricity is generated during the coasting phase. It differs from the well-known apparatus flywheel as such the latter is a storage device and not a device for electric generation. In the present invention the coasting wheel is discharging its rotational kinetic energy into electric energy instantaneously during its spinning phase when the source of energy is turned off as well as it generates electricity during the phase of energy-in. 
     The present invention includes a unique method of rotation induced by thruster engines (converging diverging duct) suited in a rectangular wing for levitation. Normally, the electric generator or alternator operates with a piston engine or turbine.

CROSS-REFERENCE TO RELATED APPLICATIONS

Not Applicable

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

Individual Efforts

BACKGROUND

It may be argued that the function of the present invention is related to the function of the conventional electric generator and the principle of the flywheel. However, a typical electrical generator/alternator is an apparatus which converts mechanical energy (or power) into electrical energy (or power). It is based on the principle of production of dynamic induced emf (Electromotive Force). This emf causes a current to flow if the conductor circuit is closed. The key different between an alternator and a generator is what spins and what is fixed. The windings of wire (the armature) spin inside a fixed magnetic field in the case of generator. In the case of an alternator, a magnetic field is allowed to spin inside of windings of wire called a stator to generate the electricity.

First, up to present, all electric generators and alternators include an element of stator and an element of rotor but in the present invention both the armature and magnets are rotors. Second, all electric generators operate in non-harmonic mode, i.e., when the input fuel is turned off the electric generator or alternator stops. Third, the present invention includes a unique method of rotation induced by thruster engines (converging diverging duct) suited in a rectangular wing for levitation. Normally, the electric generator or alternator operates with a piston engine or turbine.

A flywheel is an energy storage device that uses its significant moment of inertia to store energy as a rotational kinetic, and then deliver it to a load at the appropriate time, in the form that meets the load needs. The faster the flywheel spins the more kinetic energy it stores, and the heavier is the mass of the disc the more kinetic energy it stores. A flywheel is storage device and not a device for electric generation. In the present invention, the coasting wheel is discharging its rotational kinetic energy into electric energy instantaneously during its spinning phase when the source of energy is turned off as well as it generates electricity during the phase of energy-in.

BRIEF DESCRIPTION OF DRAWINGS

Figures included in this invention are briefly described as follows.

FIG. 1 An overall description of the coasting-wheel with aeronautic levitation for harmonic electric generation. All elements are shown with alphabetic letters. The components and subcomponents are indicated by their numbers in the consequence figures.

FIG. 2 The cylindrical vessel of the present invention. It is a stationary vessel for holding and protecting the components of the generator.

FIG. 3 The feeding-pressure system.

FIG. 4 The upper thruster-and-levitation system. It includes a pre-entry cylinder and a rotary cylinder. The latter includes the basic elements of the thrusters and levitation.

FIG. 5 An expanded view of the pre-entry cylinder of the upper thruster-and-levitation system.

FIG. 6 The position of the thruster is shown with respect to the combustion chamber.

FIG. 7 The geometric orientation of the wings and thruster for the upper thruster-and-levitation system.

FIG. 8 The compartment of coasting wheels is briefly illustrated. It comprises the following basic units: the upper coasting, the EM (electromagnetic induction) elements; the armature coil and permanent magnets of different poles, and the lower coasting wheel

FIG. 9 The detailed components of the upper coasting wheel.

FIG. 10 The detailed components of the lower coasting wheel.

FIG. 11 The lower thruster-and-levitation system. It includes a pre-entry cylinder and a rotary cylinder. The latter includes the basic elements of the thrusters and levitation.

FIG. 12 An expanded view of the pre-entry cylinder of the lower thruster-and-levitation system.

FIG. 13 The geometric orientation of the wings and thruster for the lower thruster-and-levitation system.

FIG. 14 The setup motion of the conductor (coil) and permanent magnet. The conductor wires are not shown.

FIG. 15 The stationary configuration of the coasting-wheel. This is valid for both coasting wheels.

FIG. 16 The rotational configuration of the coasting wheel. This is valid for both coasting wheels.

FIG. 17 Illustration of the centripetal force and centrifugal force acting on a rotating wheel.

FIG. 18 Illustration of the simple harmonic oscillation. The oscillation is shown with damping.

FIG. 19 Characterization of the thruster with thermodynamic variables.

FIG. 20 Illustration of the moment of force (torque) induced by the thruster.

SUMMARY OF THE INVENTION

The present invention titled: a coasting-wheel with aeronautic levitation for harmonic electric generation is a device to generate electric energy or power during the phase of energy-in and during the coasting phase; energy-off. The present generator holds a promising uses for terrestrial and space domains.

One embodiment of the present invention is the dual motions of the conductor (armature) and permanent magnets during energy in and off. Two coasting wheels are used in the present invention. One carries the armature and the second carries the permanent magnets.

A second embodiment is the aeronautics levitation of the coasting wheel with rectangular wings. Each coasting wheel is levitated with a pair of wings oriented in the reverse direction in which the leading edge of the air foil indicates to the direction of motion.

A third embodiment of the present invention is the use of thrusters (converging diverging nozzle) to propel the costing wheel. Each coasting wheel is propelled by two thrusters oriented on the reverse directions fixed inside the wing. In this case, the direction of the thrust is at the same direction of the leading edge of the of the air foil of the wing.

A fourth embodiment is the applicability of using different types of energy sources for the kickoff motion this includes compressed gas binary or singular, liquid fuel and oxidizer, and gaseous fuel and oxidizer.

The coasting-wheel with aeronautic levitation for harmonic electric generation comprises the following elements as indicated by their labels in the brief diagram shown in FIG. 1. A cylindrical vessel (A) contains the inner components of the generator and provides a protection to the inner components. A feeding-pressure system (B) comprises a group of pressure regulators, feeding tubes of compressed gas or liquids. The upper thruster-and-levitation system (C) comprises a pre-entry stationary cylinder and a rotary entry cylinder, a pair of rectangular wing, a pair of converging diverging thrusters, and a system of feeding fuel. The compartment of coasting wheels (D) comprises the following basic units: the upper coasting, the EM (electromagnetic induction) elements; the armature coil and permanent magnets of different poles, and the lower coasting wheel. The lower thruster-and-levitation system (E) comprises a pre-entry stationary cylinder and a rotary entry cylinder, a pair of rectangular wing, a pair of converging diverging thrusters, and a system of feeding fuel. The upper and lower coasting wheels are allowed to rotate in the opposite direction.

DETAILED DESCRIPTION OF THE INVENTION I) The Structure of the Invention

The present invention which is titled a coasting-wheel with aeronautic levitation for harmonic electric generation is an apparatus of converting the energy from compressed gas or fuel or fuel and oxidizer (liquid or gaseous phases) into rotational mechanical energy through expansion of the compressed gas or burning the fuel in converging diverging duct (thruster). The generated rotational mechanical energy is therefore converted into electric energy via the principle of electromagnetic induction. The apparatus is oriented vertically (aloft), consequently; the coasting wheels and the elements of the electric generators suffer a rotation in which the rotational plane is parallel to the normal of the gravitational force (=mg). In the present invention, the full cycle of the rotation of the wheel is accomplished against the friction force (=μmg), where μ is the coefficient of friction. Thus by selecting a material with least coefficient of friction (e.g. zinc oxide ZnO, or diamond and graphite; μ=0.1 without lubricant) a harmonic motion can be sustained for a longer of time.

In common electric generators, the rotational plane is perpendicular to the normal of the gravitational force (=mg). In this case, half of the cycle requires a kinetic energy greater than the potential energy (=−mgh), where h is the radius of the rotational plane, and the second half of the cycle is accomplished by the gravitational force mg (=+mgh). However this type of rotation cannot be used for harmonic motion since it decays faster when the source of energy is turned off, because of the negative potential energy and friction.

The present apparatus comprises the following components as indicated by the alphabetic letters and consequent numbers.

I-1) The Cylindrical Vessel (A)

The cylindrical vessel (1) comprises the following elements, as shown in FIG. 2. A cylinder head (2) which contains four feeding tubes as indicated in part (B17, B18, B19, B20). A tube holder (3) configured cylindrically, two placed in the upper portion (nearer to the head of the vessel) of the vessel and two placed at the lower portion (nearer to the bottom of the vessel). These holders are used to keep the feeding tubes (B19 and B20) motionless. A wall aperture (4) is for the purpose to exhaust the exit gas from the thrusters to the atmosphere or to the ambient in general. Two wall apertures are located in the upper portion and two wall apertures are located at the bottom portion of the vessel. An upper stationary tray (5), located above the central height in order to fix the upper coasting wheel (Dl) from its center. A lower stationary tray (6), located below the central height in order to fix the lower coasting wheel (D10) from its center.

I-2) The Feeding-Pressure System (B)

The present electric generator is operated by two coasting wheels (upper and lower), each coasting wheel is propelled by two thrusters, and each thruster is fed by a pair of tubes. The feeding-pressure system comprises the following elements, as shown in FIG. 3. An inlet tube of pressurized gas or liquid (1), a pressure valve (2), a pressure counter (3), a tube (4) fixed to tube (19) which is connected to the lower thruster-and-levitation system. An inlet tube of pressurized gas or liquid (5), a pressure valve (6), a pressure counter (7), a tube (8) fixed to tube (17) which is connected to the upper thruster-and-levitation system. An inlet tube of pressurized gas or liquid (13), a pressure valve (14), a pressure counter (15), a tube (16) fixed to tube (18) which is connected to the upper thruster-and-levitation system. An inlet tube of pressurized gas or liquid (9), a pressure valve (10), a pressure counter (11), a tube (12) fixed to tube (20) which is connected to the lower thruster-and-levitation system.

I-3) The Upper Thruster-and-Levitation System (C)

The upper thruster-and-levitation system (C) comprises the following elements, as shown in FIG. 4. From top view, it includes a pre-entry stationary cylinder unit (1) and a rotary entry cylinder (3). Further the pre-entry cylinder (1) is filled with several solid rods (2). The purpose of the solid rods is to minimize volume of the space of element 1 so that the gas or liquid pours from tube B18 inside element (1) with no expansion. The extension of the tube B17 is continuing inside the element (1) as shown in the figure. The tube B18 is fixed to the surface of the element (1). The bottom (far from entry side) of the pre-entry cylinder includes a radial bearing (mechanical or magnetic) with closed surface, unit 5. Further, unit 5 includes a smaller radial bearing (mechanical or magnetic) located at the center, unit 6 and aperture unit 7 located away from center. The purpose of the aperture (unit 7) is to let the incoming gas or liquid poured by tube B18 into unit 3.

The extension of tube B18 is fixed to the inner ring of the radial bearing, unit 6. Further, the rotary entry cylinder, unit 3, has a head (unit 4). The head (4) is fixed to the inner ring of the radial bearing unit 5. Thus, during rotation unit 1 is motionless except at bearing section. An adapter, unit 8, a tube fixed to the outer ring of the radial bearing unit 6 as shown of the expanded view in FIG. 5. Two channels are emerging from unit 1; tube 9 and tube 10. Tube 9 is fixed to the adapter (8) and passes the incoming gas or liquid from B17. Wherein the adapter is to match the inner diameter of B17 and tube 9, thus the flow moves without any expansion. The purpose of tube 10 is to intake the gas or liquid poured by B18 during rotation. The moving parts at this phase are elements 8, 9, and 10. Channel 9 is fixed to an inverted-T adapter (unit 11), to feed the thruster units to the right and left of the diagram. Channel 10 is fixed to an inverted-T adapter (unit 12) to feed the thruster units to the right and left of the diagram. Further, unit 3 includes two rectangular wings fixed at its center.

The left wing (unit 13), is oriented so that the leading edge of the air foil (14) indicates for the direction of motion (thrust motion 18). At the entry of the left wing (13), a combustion chamber (15) is suited inside with two inlets of gas or liquids as shown in FIG. 6. A thruster (element 16), a converging diverging duct with a rectangular cross section is fixed to the combustion chamber from the converging part. Thus the direction of the outlet gas (17) is from the diverging part, and the thrust (18) indicates for the direction of motion.

The right wing (unit 19), is oriented in the reverse direction of unit 13 so that the leading edge of the air foil (20) indicates for the direction of motion (thrust direction 24). At the entry of the right wing (19), a combustion chamber (21) is suited inside with two inlets of gas or liquids as shown in FIG. 6. A thruster (element 22), a converging diverging duct with a rectangular cross section is oriented reversely and fixed to the combustion chamber from the converging part. Thus the direction of outlet gas (23) is from the diverging part, and the thrust (24) indicates for the direction of motion. As shown in the Figure, the motion is counterclockwise (26). The kinetic rotational energy is transferred to the upper coasting wheel via the shaft (element 25).

The rotation is caused because both thrusters are oriented tangentially to unit 3, the thrust is perpendicular to the line of axis that is extended from the center and passes to the end of both wings as shown in FIG. 7. The upper thruster-and-levitation system unit causes a rotation counterclockwise (26) as shown in the diagrams.

I-4) The Compartment of Coasting Wheels (D)

The compartment of coasting wheels comprises the following basic units: the upper coasting wheel, the EM (electromagnetic induction) elements; the armature coil and permanent magnets of different poles, and the lower coasting wheel as shown in FIG. 8. While the upper coasting wheel (1) is fixed from its center in tray A5, the lower coasting wheel (10) is fixed from its center in tray A6. The armature (coil-conductor), unit 7, is fixed to the bottom of the upper coasting wheel by a shaft (2). The electric energy (or power) is extracted from the armature (7) by a pair of electric conducting rods (3) and brushes 4. Further, the brushes (4) are attached to a flexible conducting wire (5), and the latter are attached to electric terminals (6) which are fixed on top of tray A5. The permanent magnets of different poles (8) are fixed on top of the tray 9 which is fixed on top of the lower coasting wheel (unit 10). The inner unoccupied volume of unit D is air free (vacuum).

The detailed components of the upper coasting wheel are shown in FIG. 9. It comprises a hollow cylindrical shaped cage (1) of which a wheel (11) is placed at the central position. The wheel can be made of any rigid material (such as stainless steel) but its surface (18) is coated or made from least friction material such as zinc oxide ZnO which has similar properties of diamond. Zinc oxide single crystals are grown (synthesized) by the modified vertical Bridgman method. The Bulk Young's modulus of ZnO is 111.2 GPa. The extension of element C25, a shaft, is extended through inside the cage with a clearance (13) from the inner wall of the cage, further the shaft C25 is fixed to the top of the coasting wheel. The cage, unit 1, is motionless. The wheel (11) is allowed to slide within two types of rolling balls at the off-axial and radial directions. While the radial rolling balls are shown by number 19, the off-axial (up and down) rolling balls are shown by numbers 17 and 20 respectively. The rolling balls 17 and 20 are off the center axis because of the location of the shaft. All rolling balls are enclosed in a cage (16) similar to cages used in mechanical bearings. The cages (16) of the off-axial rolling balls are mounted on helical springs (15) and fixed to the inner top and bottom surfaces (14) of unit 1. The lower shaft (12) of the wheel is the extension of the shaft 2, however; both are shown in different diameters.

The detailed components of the lower coasting wheel are shown in FIG. 10. It comprises equivalent components, materials, and dimension, to the upper coasting wheel. The cage of the wheel is denoted by 10. The repeated numbers in the diagram are referred in the previous section. The upper shaft (21) carries the tray (9), and the lower shaft of the coasting wheel is the extension of the shaft E25. The upper and lower coasting wheels are rotating in the opposite direction.

Even though the number of rolling balls is shown two pieces in each direction for each coasting wheel but however a good stability of the wheel during motion will be achieved with additional numbers particularly in the off-axial direction. The purpose of using radial rolling balls is to keep the position of the wheel fixed at the central axis while undergoing a rotation. The use of magnetic bearing in the radial direction to stabilize the coasting wheel during rotation is frictionless but is more expensive. The use of magnetic bearing in the off-axial direction is more complex since the wheel will be levitated up during motion and thus the presence of a mechanical spring is vital in this case.

I-5) The Lower Thruster-and-Levitation System (E)

The lower thruster-and-levitation system (E) comprises the following elements, as shown in FIG. 11. It includes equivalent materials, dimension and components to the upper thruster-and-levitation system but the orientations of the components are reversely and thus a different numbers are cited. From bottom view, it includes a pre-entry stationary cylinder unit (1) and a rotary entry cylinder (3) on top of the pre-entry stationary cylinder. Further the pre-entry cylinder (1) is filled with several solid rods (2). The purpose of the solid rods is to minimize volume of the space of element 1 so that the gas or liquid from connection B19 pours inside element (1) with no expansion. The extension of the tube B20 is continuing inside the element (1) as shown in the figure. The tube B19 is fixed to the surface of the element (1). The top (far from entry side) of the pre-entry cylinder includes a radial bearing (mechanical or magnetic) with closed surface, unit 5. Further, unit 5 includes a smaller radial bearing (mechanical or magnetic) located at the center, unit 6 and aperture unit 7 located away from center. The purpose of the aperture (unit 7) is to let the incoming gas or liquid poured by tube B19 into unit 3.

The extension of tube B20 is fixed to the inner ring of the radial bearing, unit 6. Further, the rotary entry cylinder, unit 3, has a head (unit 4). The head (4) is fixed to the inner ring of the radial bearing unit 5. Thus, during rotation unit 1 is motionless except at bearing section. An adapter, unit 8, a tube fixed to the outer ring of the radial bearing unit 6 as shown of the expanded view in FIG. 12. Two channels are emerging from unit 1; tube 9 and tube 10. Tube 9 is fixed to the adapter (8) and passes the incoming gas or liquid from B20. Wherein the adapter is to match the inner diameter of B20 and tube 9, thus the flow moves without any expansion. The purpose of tube 10 is to intake the gas or liquid poured by B19. The moving parts at this phase are elements 8, 9, and 10. Channel 9 is fixed to an inverted-T adapter (unit 11), to feed the thruster units to the right and left of the diagram. Channel 10 is fixed to an inverted-T adapter (unit 12) to feed the thruster units to the right and left of the diagram. Further, unit 3 includes two rectangular wings fixed at its center.

The right wing (unit 13), is oriented so that the leading edge of the air foil (14) indicates for the direction of motion (thrust direction 18). In this case, unit 13 is in the reverse direction of unit C19. At the entry of the right wing (13), a combustion chamber (15) is suited inside with two inlets of gas or liquids as shown in FIG. 6. A thruster (element 16), a converging diverging duct with a rectangular cross section is fixed to the combustion chamber from the converging part. Thus the direction of the outlet gas (17) is from the diverging part, and the thrust (18) indicates for the direction of motion. In this case the orientation of the thruster unit 16 is in the reverse direction of C22.

The left wing (unit 19), is oriented in the reverse direction of unit 13 so that the leading edge of the air foil (20) indicates for the direction of motion (thrust direction 24). In addition, unit 19 is also in the reverse direction of unit C13. At the entry of the left wing (19), a combustion chamber (21) is suited inside with two inlets of gas or liquids as shown in FIG. 6. A thruster (element 22), a converging diverging duct with a rectangular cross section is oriented in the reverse direction of unit 16 and fixed to the combustion chamber from the converging part. Thus the direction of outlet gas (23) is from the diverging part, and the thrust (24) indicates for the direction of motion. As shown in the Figure, the motion is clockwise (26). The kinetic rotational energy is transferred to the lower coasting wheel via the shaft (element 25).

The rotation is caused because both thrusters are oriented tangentially to unit 3, the thrust is perpendicular to the line of axis that is extended from the center and passes to the end of both wings as shown in FIG. 13. The lower thruster-and-levitation system unit causes a rotation clockwise (26) as shown in the diagrams.

All wings that are presented in the upper and lower thruster-and-levitation systems are of rectangular shape. A rectangular wing with sharp end is adopted in the present invention because it can carry a reasonable load and levitate at a reasonable speed. It is inexpensive to build and maintain.

II) The Concept of the Invention

A coasting-wheel oscillator is an apparatus which converts mechanical energy (or power) from thruster engine into electrical energy (or power). The electric energy is produced during engine on, kickoff mode, and during engine off, coasting mode. The term: “coasting” refers to the inertial motion of the object according to Newton's First Law, when the engine shuts off. The electric part of the invention is based on the principle of production of dynamic induced electromotive force. Basically, whenever a conductor cuts magnetic flux, dynamically induced electromotive force ε (volt) is produced in it according to Faraday's Laws of Electromagnetic Induction:

$\begin{matrix} {ɛ = {{- N}{\frac{\varphi}{t}.}}} & {{II}{.1}} \end{matrix}$

Where: N is the number of turns of the conductor (coil), Δφ is the magnetic flux in Tesla-m² and Δt is the time of induced motion. This ε causes a current to flow if the conductor circuit is closed. In the present invention, the conductor and the magnets are allowed to move in the opposite direction, i.e. both are rotors, as shown in FIG. 14. If we consider, an equal rotational speed v, then 8 in this case can be written as

$\begin{matrix} {ɛ = {{- 2}{\left( {N\frac{\varphi}{t}} \right).}}} & {{II}{.2}} \end{matrix}$

The factor of 2 is inserted in Equation II-2 because both the conductor and magnets move in the opposite direction at the same speed. Further, the electromotive force for when the conductor and the magnet suffer a rotational motion in the opposite direction can be written as

ε=2N BA sin ωt.  II.3

Where: A is the projected area of the conductor with the magnetic field B (Tesla).

If comparison is made between the present invention and a typical electric generator or alternator, the features of the present invention, without the considerations of the harmonic motion of the coasting-wheel are:

-   -   1—For equal input energy, the electromotive force is the same,         but the speed of the shafts of the coasting wheels is one half         of the speed of the shaft of the electric generator or         alternator.     -   2—For equal shaft speed, the electromotive force is twice than         the case of electric generator or alternator.     -   In both cases, it is assumed that the number of turns of         conductor, the shape of the conductor, and the magnetic flux are         kept constants.

The mechanical energy is generated by rocket engines (convergence-divergence duct) placed on wings with opposite arrangement; as a result a rotational thrust is produced. The mechanical (rotational) kinetic energy generated by the engine is transferred through a shaft to the coasting wheel which is located in a separate compartment. Two coasting wheels are used in this invention, one carries the conductor (the coil) and the other carries the permanent magnets, both are revolving in the opposite direction. The coasting wheel is supported by radial rolling balls (which acts as a radial bearing) and axial rolling ball attached to helical spring.

When the coasting wheel is in stationary mode, FIG. 15, it sags on the rolling balls which are located at the bottom of the unit. In this case, the helical springs of the bottom rolling balls are in compressed state. At the same time the upper rolling balls are just touching the surface of the wheel, and their helical springs are in normal (neither compressed nor stretched) condition. Therefore the lower springs holds the weight (=m g) of all elements attached to the coasting wheel.

When the coasting wheel is in rotation, FIG. 16, and because of the air-foil wings it levitates such that; the surface of the wheel will begin to compress the helical springs of the rolling balls located at the upper part of the unit. At the same moment, the helical springs of the rolling balls located at the bottom of the unit begin to relief their compressed state. The maximum relief of the helical springs of the bottom rolling balls reaches the normal state of the springs, at this state; the upper rolling balls reach a maximum compression state. The role of the radial rolling balls is to align the wheel in the axial position. Thus the wheel is allowed to slide on upper and lower rolling balls in order to maintain its inertia. For when the wheel maintains its motion after the engine shuts off, the wheel is said to be in inertial motion. Inertia is the property of an object which describes its tendency to remain at rest or in uniform motion according to Newton's First Law. The concept of inertia also surfaces in the discussion of any problem involving rotation about an axis with the quantity called the moment of inertia being a measure of how an object resists a change in its rotation.

When an object maintains a periodic motion after kickoff energy, the motion is said to be in harmonic state at which the initial conditions force and energy are restored for a constant period. Therefore, in circular motion, inertia and harmonic lead to the same conclusion. Thus the coasting wheel can be considered in a harmonic motion for a period of time when the engine shuts off because of inertia.

In an ideal oscillator, the harmonic motion can be sustained for a longer time (infinity), however because of resistive force originated from air resistance, friction and material stress, the harmonic motion is damped, that is; the amplitude (energy) becomes smaller than the initial amplitude. The initial velocity v of the coasting-wheel is

$\begin{matrix} {v = {\sqrt{\frac{2\mspace{14mu} E}{m}}.}} & {{II}{.4}} \end{matrix}$

Where E is the input energy and m is the mass of the wheel and its attachments. In order to increase the rotational speed one can increase the input energy or reduce the mass of the system. It is must be noted that we concern about the speed of the coasting wheel and not the kinetic energy of the coasting wheel.

II-1) The Harmonic Motion of the Coasting Wheel

In general words, simple harmonic motion is the motion of an object or system that is subject to a restoring force or energy for a given constant time or frequency. If we consider a wheel of r dimension and m mass rotates about its axis as a result of kickoff energy or force, FIG. 17, slides on a smooth rolling balls as mentioned earlier. The wheel because of inertia maintains its motion harmonically of which two forces govern its motion exerted in the opposite direction. First, the centripetal force (toward center) F_(c), according to Newton's second law is given by:

$\begin{matrix} {F_{c} = {m{\frac{v^{2}}{r}.}}} & {{II}\text{-}1.1} \end{matrix}$

Here m is the mass of the rotating object or system and v is the rotational velocity, and r is the half diameter. In a rotation system, the velocity v is defined as the angular velocity and it is equal to co r, ω is the angular frequency in radian per second (ω is equal to 2 π/T, T is the period of the motion) and r is the orbital radius of the rotating object in meters. Then Equation II-1.1 becomes

F _(c) =mω ² r.  II-1.2

Second, the centrifugal force (away from center) F_(g), according to Newton's second law, it is equal to

$\begin{matrix} {F_{g} = {m{\frac{^{2}r}{t^{2}}.}}} & {{II}\text{-}1.3} \end{matrix}$

The second order derivative term is the radial acceleration. The motion of the wheel is balanced for when both forces are equal:

$\begin{matrix} {{m\mspace{14mu} \omega^{2}\mspace{14mu} r} = {{- m}{\frac{^{2}r}{t^{2}}.}}} & {{II}\text{-}1.4} \end{matrix}$

The minus sign in the above equation is to indicate that both forces are in the reverse direction. Rearrange Equation II-1.4, it becomes

$\begin{matrix} {{\frac{^{2}r}{t^{2}} + {\omega^{2}\mspace{14mu} r}} = 0.} & {{II}\text{-}1.5} \end{matrix}$

Equation II-1.5 is a two-order linear differential equation represents a simple harmonic oscillator without damping.

The description of simple harmonic motion assumes that the only forces acting on the system being described result in a restoring force that is proportional to the initial energy. When that is true, of course, the resulting motion is purely sinusoidal with constant amplitude—that is, the oscillator will continue to oscillate forever with undiminished energy. None of that can be correct, of course, because there will always be either some damping of the oscillation due to either imposed or inadvertent resistive forces such as air resistance, friction, material internal stress, environmental factors such as heat, etc. Therefore the real oscillation of harmonic motion is subject to damping because of the resistive force F_(r) which can be envisioned acting on the side of the centrifugal force. In most literatures, the resistive force is written as

F _(r) =−bv.  II-1.6

Where: b is a damping coefficient and v is the velocity in the radial direction it can be written as dr/dt. Therefore, Equation II-1.4 can be written as, for damping harmonic oscillator:

$\begin{matrix} {{m\mspace{14mu} \omega^{2}\mspace{14mu} r} = {{{- m}\frac{^{2}r}{t^{2}}} - {b{\frac{r}{t}.}}}} & {{II}\text{-}1.7} \end{matrix}$

After arrangement, yields

$\begin{matrix} {{\frac{^{2}r}{t^{2}} + {\frac{b}{m}\frac{r}{t}} + {\omega^{2}\mspace{14mu} r}} = 0.} & {{II}\text{-}1.8} \end{matrix}$

Equation II-1.8 is a homogeneous second-order linear differential equation represents a simple harmonic oscillator rotational motion. It has a solution of

r(t)=Ce ^(−(b/2m)t) cos(ωt).  II-1.9

Where: C is a constant.

In circular motion, present case, the harmonic oscillation is determined from the displacement incremental distance Δr as shown in Figure. For small angle;

Δr=rθ.  II-1.10

Where: θ is in radian and it can be expressed θ=ωΔt, where co is a constant (2π/T), T is the period of the motion. Then Equation II-1.9 becomes

Δv(t)=C′e ^(−(b/2m)t) cos(ωt).  II-1.11

Where: Δv=Δr/Δt, and C′=C ω (amplitude of the velocity increment). The time dependence of the amplitude is given by the decreasing exponential function—with the argument of the exponential given by b/2m. That implies that the larger the damping coefficient b in the damping force, the more quickly the oscillations diminish and the larger the mass, the less quickly the oscillations die out. However, for the latter case a tradeoff must be made since at the same time we need to keep the mass as low because of Equation II.4.

Illustration of the real simple harmonic oscillation with damping is shown in FIG. 18. The primary effect of the damping force on the motion of the oscillator—assuming the damping is not so great as to prevent it from oscillating at all (a situation called “below-damping”)—is to cause the amplitude of the oscillation to diminish in time. The asymptote of the oscillation as shown in the diagram is the exponential term of Equation II-1.11, so the amplitude of the decay of the oscillation can be written as

C(t)=C ₀ e ^(−(b/2m)t).  II-1.12

Therefore the electromotive force ε during the coasting period (From Equations II.3 and II-1.12):

ε(t)=2N BA sin ωt e ^(−(b/2m)t)  II-1.13

So, the beneficiary of operating the present electric generator in the coasting mode is obviously seen from Equation II-1.13. Such energy cannot be available by conventional electric generators. However, one more factor needs to be reduced or eliminated in order to keep the damped oscillation as long as possible and this is the reversed build-up magnetic force. Because all conductors exhibit an effective diamagnetism when they experience a changing in magnetic field an eddy current is induced. The eddy currents then produce an induced magnetic field opposite the applied field, resisting the conductor's motion.

However, dozens of technologies are currently applied in order to reduce the eddy currents in the metal armature. The most common is by letting the conductor wires to be wrapped around a number of thin metal sheets called laminations. Each lamination is insulated from the adjacent plates. The laminations disrupt any induced eddy currents, and consequently reduce the reverse build-up magnetic force.

II-2) The Propulsion of the Coasting-Wheel

Each coasting wheel (two) is driven through a shaft which is mechanically propelled via two thrusters (engine), shown converging-diverging nozzle. Each thruster is fixed in an air-foil wing which is fixed in rotary cylinder. Each rotary cylinder uses two wings and each wing is propelled by a thruster which is placed in the reverse direction of the second thruster. The wing is oriented such that the leading edge of the air foil indicates to the direction of motion. The thruster may propelled by a liquid or gaseous propellant (compressed gas or fuel and oxidizer or singular fuel).

Liquid propellant engines use liquid propellants that are fed under pressure from tanks into a thrust chamber. The liquid bipropellant consists of a liquid oxidizer (e.g., liquid oxygen) and a liquid fuel (e.g., kerosene). A monopropellant is a single liquid that contains both oxidizing and fuel species; it decomposes into hot gas when properly catalyzed (e.g., hydrogen peroxide H₂O₂). In both cases, in the thrust chamber the propellants react to form hot gases, which in turn are accelerated and ejected at a high velocity through a supersonic nozzle.

Gaseous propellant engines are classified as cold engines and hot engines. The cold engines use a stored high-pressure gas, such as air, nitrogen, or helium, as their working fluid or propellant. The high pressure-gas is fed to the system as shown before. Monopropellant and bipropellant can be applied here. The hot engines use one of the types of the liquid propellant in gaseous phase, and the high pressure gas is fed to the system from high pressure vessels.

The propellants expands or burn in the thrusters, converging-diverging nozzle and its internal energy is converted into kinetic energy and the thrust is produced by the gas pressure on the surfaces exposed to the gas. As the mass is ejected from the thruster at high speeds in the exhaust flow, thrust is produced in the opposite direction according to Newton's third law. Thrust is also produced by any difference between the exhaust pressure and the ambient pressure. In a converging-diverging nozzle a large fraction of the thermal energy of the gases in the chamber is converted into kinetic energy, under the assumption of the isentropic expansion of the flow in the duct, the thrust can be found

F _(Th) ={dot over (m)}v _(e)+(P _(e) −P _(a))A _(e).  II-2.1

Where F_(Th) is the thrust in N, {dot over (m)} is the mass flow rate in kg/s, v_(e) is the exist velocity in m/s, P_(e) is the exit pressure in Pascal, P_(a) is the ambient pressure in Pascal, and A_(e) is the exit area of the duct in m² as shown in FIG. 19. The letters A, p, v, T, and h indicate for the cross sectional area, pressure, velocity, temperature, and stagnation enthalpy respectively. The subscripts, i, t, and e indicate for inlet, throat, and exit sectors of the converging-diverging thruster. It must be noted that the pressure at the inlet is evaluated at the chamber pressure (the pressure of the compressed vessels that are located outside the whole generator). All variables in Equation II-2.1 and the design of the thrusters can be found from well-known thermodynamic relations.

Each thruster induces a moment of force on the shaft of the rotary cylinder holding the coasting wheel, i.e. torque τ according to the relation in scalar product (FIG. 20):

τ=d·F·sin α.  II-2.2

Where: d is the arm (distance) of the thrust force acting on the center of the shaft and α is the angle of which the force makes with the arm. Because the thruster is placed on the tangent of the circle (rotary cylinder) the exit gas, consequently the thruster, makes 90 degrees. Hence Equation II-2.2 becomes

τ=d·F.  II-2.3

For when d is in meters and F in Newton, the moment of force is in joules (N·m). Because of each wheel is propelled by two thrusters therefore each coasting wheel is driven by two forces acting on the opposite direction but at different points of action, then the total moment of force τ_(t) is equal to

τ_(t)=2d·F.  II-2.4

Then from Equation II-2.1, the total moment of force is equal to

τ_(t)=2d({dot over (m)}v _(e)+(P _(e) −P _(a))A _(e)).  II-2.5

The total moment of force is the kinetic energy which propels the coasting wheel. Hence the velocity of the coasting wheel is

$\begin{matrix} {v = {2\sqrt{\frac{d\left( {{\overset{.}{m}v_{e}} + {\left( {P_{e} - P_{a}} \right)A_{e}}} \right)}{m}}}} & {{II}\text{-}2.6} \end{matrix}$

Therefore to achieve a higher velocity of coasting wheel, one can increase the mass flow rate, or the exit velocity, or the exit pressure, or the exit area. The mass of the coasting-wheel system can be kept high in order to allow a longer time of damping oscillation, but as mentioned they will be a compromise between designing a heavier wheel to achieve a longer oscillation time and a lighter wheel to achieve a high rotational velocity and thus a high electric energy. 

What is claimed is:
 1. An apparatus for electric power generation using a generator: a coasting-wheel with aeronautic levitation for harmonic electric generation, for terrestrial application wherein the generator comprises: a cylindrical vessel; a feeding-pressure system; an upper thruster-and-levitation system; a compartment of coasting wheels; a lower thruster-and-levitation system, further, the cylindrical vessel comprises the following elements: vessel head (points vertically), four feeding tubes, four tube holders; two placed in the upper portion of the vessel and two placed at the lower portion of the vessel, four wall apertures; two wall apertures are located in the upper portion and two wall apertures are located at the bottom portion of the vessel, an upper stationary tray located above the central height of the cylinder of which the upper coasting wheel (nearer to vessel head) is fixed from its center, a lower stationary tray located below the central height of the cylinder of which the lower coasting wheel (away from vessel head) is fixed from its center, further the unoccupied volume between the two trays is air free (vacuum), further, a feeding-pressure system comprising the following elements: The feeding-pressure system comprises the following elements; four inlet feeding tubes of pressurized fuel gas or liquid; two inlet feeding tubes for the upper thruster-and-levitation system and two inlet feeding tubes for the lower thruster-and-levitation system, further, the upper thruster-and-levitation system (nearer to vessel head) comprising the following elements: a pre-entry stationary cylinder unit (nearer to the vessel head); a rotary entry cylinder at the bottom of the pre-entry stationary cylinder, further it includes a head (nearer to vessel head) and shaft (farther from vessel head); two rectangular wings, fixed at the center of the rotatory cylinder, further the pre-entry stationary cylinder comprises at its bottom (farther from the vessel head) a radial bearing (closed) with a dimension of the cylinder diameter, further the head of the rotary cylinder is fixed to the inner ring of the radial bearing, wherein at the center of the pre-entry cylinder a smaller radial bearing with a dimension of the diameter of the feeding tube is positioned, and aperture located away from the central axis, further; one feeding tube is fixed away from the center at the upper surface (nearer to the vessel head) of the pre-entry cylinder such that the exit gas or liquid is extracted from the aperture by a tube passes through the rotary cylinder and splits by an inverted-T connection into right and left wings, the second feeding tube is passing through the center of the pre-entry cylinder and fixed at the inner ring of the smaller radial bearing such that the exit gas or liquid is extracted by an adaptor fixed at the outer ring of the smaller radial bearing, wherein at the end of the adaptor a tube passes through the rotary cylinder and splits by an inverted-T connection into right and left wings, further; each wing includes a combustion chamber at the inlet of the wing and a fixed thruster at the center of the combustion chamber, the thruster (converging diverging duct with a rectangular cross section) is placed tangent to the axis of the rotary cylinder, further, with respect to a reference of frame, the right wing is oriented such that the leading edge of its air foil indicates for the direction of the thrust, with respect to the same reference of frame, the left wing is oriented such that the leading edge of its air foil indicates for the direction of thrust but in the reverse direction of the right wing, further a compartment of coasting wheels comprising the following elements: the upper coasting wheel; the EM (electromagnetic induction) elements; the armature coil and permanent magnets of different poles; the lower coasting wheel, further, the upper coasting wheel (nearer to vessel head) comprises: a hollow cylindrical shaped cage of which a wheel is placed at the central position, the wheel can be made of any rigid material (such as stainless steel) but its surface is coated or made from least friction material such as zinc oxide ZnO which has similar properties of diamond, further the shaft from the rotary entry cylinder of the upper thruster-and-levitation system is inserted through the hollow cylindrical cage with a clearance from the inner wall of the cage, further the shaft is fixed to the top of the upper coasting wheel, further the hollow cylindrical cage is motionless, further the wheel is allowed to slide within two types of rolling balls at the off-axial direction (far from the center) with minimum four rolling balls on top and bottom of the wheel, and at the radial direction with minimum two rolling balls, all rolling balls are enclosed in a cage similar to cages used in mechanical bearings, further the cages of the off-axial rolling balls are mounted on helical springs and fixed to the inner top and bottom surfaces of the hollow cylindrical cage, the upper coasting wheel is propelled by the shaft of the rotary entry cylinder of the upper thruster-and-levitation system which is fixed on the top of the wheel (the closer side from the vessel head) and transfers the rotational kinetic energy to the armature (conductor coil) via a shaft located at its bottom (the far side from the vessel head), the electric energy (or power) is extracted from the armature by a pair of electric conducting rods and brushes; the brushes are attached to a flexible conducting wire and the latter are attached to electric terminals which are fixed on top of the upper tray (nearer to vessel head), further, the lower coasting wheel (farther from vessel head) comprises: equivalent components, materials, and dimension, to the upper coasting wheel, its upper shaft (nearer to vessel head) carries the permanent magnets with the support of tray fixed on the shaft, further its lower shaft is propelled by the rotary cylinder of the lower thruster-and-levitation system, further, the lower thruster-and-levitation system (farther from vessel head) comprising the following elements: a pre-entry stationary cylinder unit (nearer to the vessel bottom); a rotary entry cylinder at the top of the pre-entry stationary cylinder, further it includes a head (nearer to vessel bottom) and shaft (farther from vessel bottom); two rectangular wings, fixed at the center of the rotatory cylinder, further the pre-entry stationary cylinder comprises at its top (farther side from the vessel bottom) a radial bearing (closed) with a dimension of the cylinder diameter, further the head of the rotary cylinder is fixed to the inner ring of the radial bearing, wherein at the center a smaller radial bearing with a dimension of the diameter of the feeding tube is positioned, and aperture located away from the central axis, further; one feeding tube is fixed away from the center at the lower surface (nearer to the vessel bottom) of the pre-entry cylinder such that the exit gas or liquid is extracted from the aperture by a tube passes through the rotary cylinder and splits by an inverted-T connection into right and left wings, the second feeding tube is passing through the center of the pre-entry cylinder and fixed at the inner ring of the smaller radial bearing such that the exit gas or liquid is extracted by an adaptor fixed at the outer ring of the smaller radial bearing, wherein at the end of the adaptor a tube passes through the rotary cylinder and splits by an inverted-T connection into right and left wings, further; each wing includes a combustion chamber at the inlet of the wing and a fixed thruster (converging diverging duct with a rectangular cross section) at the center of the combustion chamber, the thruster is placed tangent to the axis of the rotary cylinder, further, with respect to the same reference of frame of the top unit, the right wing is oriented such that the leading edge of its air foil is in the reverse direction of the right wing of the upper-thruster-and levitation unit, similarly with respect to the same reference of frame, the left wing is oriented such that the leading edge of its air foil is in the reverse direction of the left wing of the upper-thruster-and levitation unit.
 2. The apparatus of claim 1, wherein the armature (coil conductor) and permanent magnets undergo simultaneous rotation in the opposite direction during the phase of energy-in.
 3. The apparatus of claim 1, wherein the armature (coil conductor) and permanent magnets undergo simultaneous rotation in the opposite direction during the phase of energy-off (during the coasting phase).
 4. An apparatus for electric power generation using a generator: a coasting-wheel with aeronautic levitation for harmonic electric generation, for space application wherein the generator comprises: a cylindrical vessel; a feeding-pressure system; an upper thruster-and-levitation system; a compartment of coasting wheels; a lower thruster-and-levitation system, further, the cylindrical vessel comprises the following elements: vessel head (points vertically), four feeding tubes, four tube holders; two placed in the upper portion of the vessel and two placed at the lower portion of the vessel, four wall apertures; two wall apertures are located in the upper portion and two wall apertures are located at the bottom portion of the vessel, an upper stationary tray located above the central height of the cylinder of which the upper coasting wheel (nearer to vessel head) is fixed from its center, a lower stationary tray located below the central height of the cylinder of which the lower coasting wheel (away from vessel head) is fixed from its center, further the unoccupied volume between the two trays is air free (vacuum), further, a feeding-pressure system comprising the following elements: The feeding-pressure system comprises the following elements; four inlet feeding tubes of pressurized fuel gas or liquid; two inlet feeding tubes for the upper thruster-and-levitation system and two inlet feeding tubes for the lower thruster-and-levitation system, further, the upper thruster-and-levitation system (nearer to vessel head) comprising the following elements: a pre-entry stationary cylinder unit (nearer to the vessel head); a rotary entry cylinder at the bottom of the pre-entry stationary cylinder, further it includes a head (nearer to vessel head) and shaft (farther from vessel head); two rectangular wings, fixed at the center of the rotatory cylinder, further the pre-entry stationary cylinder comprises at its bottom (farther from the vessel head) a radial bearing (closed) with a dimension of the cylinder diameter, further the head of the rotary cylinder is fixed to the inner ring of the radial bearing, wherein at the center of the pre-entry cylinder a smaller radial bearing with a dimension of the diameter of the feeding tube is positioned, and aperture located away from the central axis, further; one feeding tube is fixed away from the center at the upper surface (nearer to the vessel head) of the pre-entry cylinder such that the exit gas or liquid is extracted from the aperture by a tube passes through the rotary cylinder and splits by an inverted-T connection into right and left wings, the second feeding tube is passing through the center of the pre-entry cylinder and fixed at the inner ring of the smaller radial bearing such that the exit gas or liquid is extracted by an adaptor fixed at the outer ring of the smaller radial bearing, wherein at the end of the adaptor a tube passes through the rotary cylinder and splits by an inverted-T connection into right and left wings, further; each wing includes a combustion chamber at the inlet of the wing and a fixed thruster at the center of the combustion chamber, the thruster (converging diverging duct with a rectangular cross section) is placed tangent to the axis of the rotary cylinder, further, with respect to a reference of frame, the right wing is oriented such that the leading edge of its air foil indicates for the direction of the thrust, with respect to the same reference of frame, the left wing is oriented such that the leading edge of its air foil indicates for the direction of thrust but in the reverse direction of the right wing, further a compartment of coasting wheels comprising the following elements: the upper coasting wheel; the EM (electromagnetic induction) elements; the armature coil and permanent magnets of different poles; the lower coasting wheel, further, the upper coasting wheel (nearer to vessel head) comprises: a hollow cylindrical shaped cage of which a wheel is placed at the central position, the wheel can be made of any rigid material (such as stainless steel) but its surface is coated or made from least friction material such as zinc oxide ZnO which has similar properties of diamond, further the shaft from the rotary entry cylinder of the upper thruster-and-levitation system is inserted through the hollow cylindrical cage with a clearance from the inner wall of the cage, further the shaft is fixed to the top of the upper coasting wheel, further the hollow cylindrical cage is motionless, further the wheel is allowed to slide within two types of rolling balls at the off-axial direction (far from the center) with minimum four rolling balls on top and bottom of the wheel, and at the radial direction with minimum two rolling balls, all rolling balls are enclosed in a cage similar to cages used in mechanical bearings, further the cages of the off-axial rolling balls are mounted on helical springs and fixed to the inner top and bottom surfaces of the hollow cylindrical cage, the upper coasting wheel is propelled by the shaft of the rotary entry cylinder of the upper thruster-and-levitation system which is fixed on the top of the wheel (the closer side from the vessel head) and transfers the rotational kinetic energy to the armature (conductor coil) via a shaft located at its bottom (the far side from the vessel head), the electric energy (or power) is extracted from the armature by a pair of electric conducting rods and brushes; the brushes are attached to a flexible conducting wire and the latter are attached to electric terminals which are fixed on top of the upper tray (nearer to vessel head), further, the lower coasting wheel (farther from vessel head) comprises: equivalent components, materials, and dimension, to the upper coasting wheel, its upper shaft (nearer to vessel head) carries the permanent magnets with the support of tray fixed on the shaft, further its lower shaft is propelled by the rotary cylinder of the lower thruster-and-levitation system, further, the lower thruster-and-levitation system (farther from vessel head) comprising the following elements: a pre-entry stationary cylinder unit (nearer to the vessel bottom); a rotary entry cylinder at the top of the pre-entry stationary cylinder, further it includes a head (nearer to vessel bottom) and shaft (farther from vessel bottom); two rectangular wings, fixed at the center of the rotatory cylinder, further the pre-entry stationary cylinder comprises at its top (farther side from the vessel bottom) a radial bearing (closed) with a dimension of the cylinder diameter, further the head of the rotary cylinder is fixed to the inner ring of the radial bearing, wherein at the center a smaller radial bearing with a dimension of the diameter of the feeding tube is positioned, and aperture located away from the central axis, further; one feeding tube is fixed away from the center at the lower surface (nearer to the vessel bottom) of the pre-entry cylinder such that the exit gas or liquid is extracted from the aperture by a tube passes through the rotary cylinder and splits by an inverted-T connection into right and left wings, the second feeding tube is passing through the center of the pre-entry cylinder and fixed at the inner ring of the smaller radial bearing such that the exit gas or liquid is extracted by an adaptor fixed at the outer ring of the smaller radial bearing, wherein at the end of the adaptor a tube passes through the rotary cylinder and splits by an inverted-T connection into right and left wings, further; each wing includes a combustion chamber at the inlet of the wing and a fixed thruster (converging diverging duct with a rectangular cross section) at the center of the combustion chamber, the thruster is placed tangent to the axis of the rotary cylinder, further, with respect to the same reference of frame of the top unit, the right wing is oriented such that the leading edge of its air foil is in the reverse direction of the right wing of the upper-thruster-and levitation unit, similarly with respect to the same reference of frame, the left wing is oriented such that the leading edge of its air foil is in the reverse direction of the left wing of the upper-thruster-and levitation unit.
 5. The apparatus of claim 4, wherein the armature (coil conductor) and permanent magnets undergo simultaneous rotation in the opposite direction during the phase of energy-in.
 6. The apparatus of claim 4, wherein the armature (coil conductor) and permanent magnets undergo simultaneous rotation in the opposite direction during the phase of energy-off (during the coasting phase). 